Persistent live, 32-bit, in BIOS mode and UEFI mode even secure boot
It is possible but difficult to find an operating systems, that is 32-bit, and can boot both in BIOS mode and UEFI mode. Such a system can boot most PC computers with Intel/AMD processors.
The following link might be useful for you. It describes a persistent live Ubuntu based system with a 32-bit kernel, that can boot also in UEFI mode, even with secure boot,
LXLE is a 'respin' developed from Lubuntu which is a 'community flavour' developed from Ubuntu.
64-bit systems boot both in BIOS mode and UEFI mode
Installed system, boots both in BIOS mode and UEFI mode
It is possible to create an installed system in a USB pendrive, that can boot both in BIOS mode and UEFI mode. But this system does not work in 32-bit computers.
See this link,
An easy shortcut is to start from a compressed image file, which is described in the link.
Persistent live system, boots both in BIOS mode and UEFI mode
You can create a persistent live system from a 64-bit Ubuntu community flavour systems (Kubuntu, Lubuntu, ... Xubuntu), that works also in UEFI mode. But it will not work in computers with a 32-bit processor.
It is easier to make it as flexible as possible, when running in BIOS mode.
Several operating systems for 32-bit and 64-bit computers in BIOS mode
All 32-bit operating systems for PC computers with Intel/AMD processors boot in 32-bit computers and 64-bit computers. Most of them will not boot in UEFI mode. This is certainly an alternative, but new computers are usually delivered with Windows installed in UEFI mode, and you would need to enter the UEFI/BIOS menu system and set it in BIOS mode alias CSM alias legacy mode in order to run such a system in such new computers.
Persistent live, 32-bit, boots both in BIOS mode and UEFI mode, but not secure boot
Such Ubuntu and Ubuntu community flavour systems (Kubuntu, Lubuntu, ... Xubuntu) can be used to create a persistent live drive, that works also in UEFI mode. But they will usually not boot with secure boot. See these links,